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\begin{center}
\vskip 1cm{\LARGE\bf Higher Order Derivatives of Trigonometric
Functions, Stirling Numbers of the Second\\ \vskip .1in Kind, and
Zeon Algebra} \vskip 1cm \large Ant\^onio Francisco
Neto\footnote{This work was supported by Conselho Nacional de
Desenvolvimento Cient\'ifico e Tecnol\'ogico
(CNPq-Brazil) under grant 307617/2012-2.}\\
DEPRO, Escola de Minas\\
Campus Morro do Cruzeiro, UFOP\\
35400-000 Ouro Preto MG \\
Brazil \\
\href{mailto:antfrannet@gmail.com}{\tt antfrannet@gmail.com}\\
\end{center}

\vskip .2 in

\begin{abstract}

In this work we provide a new short proof of closed formulas for
the $n$-th derivative of the cotangent and secant functions using
simple operations in the context of the Zeon algebra. Our main
ingredients in the proof comprise a representation of the ordinary
derivative as an integration over the Zeon algebra, a
representation of the Stirling numbers of the second kind as a
Berezin integral, and a change of variables formula under Berezin
integration. The approach described here is also suitable to give
closed expressions for higher order derivatives of tangent,
cosecant and all the aforementioned functions hyperbolic
analogues.

\end{abstract}

\section{Introduction}

In this work, using basic operations on the Zeon algebra
\cite{Fein,Fein1,Scho,Scho1}, we will give a simple and short
proof of the following closed formulas for the $n$-th derivative
of the cotangent and secant functions
\cite{Adam,Boy,Cvi1,Cvi2,Cvi3}.
\begin{theorem}\label{thmcotsec} Let $n \geq 1$ be an integer. Then
\begin{equation}\label{derncot}
\frac{d^n \cot(x)}{d x^n}
=(2i)^n\bigl(\cot(x)-i\bigr)\sum_{k=1}^n\frac{k!}{2^k}S(n,k)\bigl(i\cot(x)-1\bigr)^k
\end{equation} and
\begin{equation}\label{dernsec}
\frac{d^n \sec(x)}{d x^n}
=\sec(x)i^n\sum_{j=0}^n(-1)^jj!\sum_{k=j}^n{n \choose k}S(k,j)2^{k-j}\bigl(i \tan (x)+1\bigr)^j,
\end{equation}
where $i:=\sqrt{-1}$, ${n \choose k}:=n!/\bigl(k!(n-k)!\bigr)$ and
$S(n,k)$ denotes the Stirling numbers of the second
kind.\end{theorem}

The determination of closed expressions for higher order
derivatives of trigonometric functions is a subject of recurrent
interest
\cite{Adam,Apo,Bern,Boy,Carl1,Carl2,Chan,Cvi1,Cvi2,Cvi3,Fra,Hof1,Hof,Knu,Kol,Kri,Shi}.
As remarked earlier \cite{Cvi1,Cvi2,Cvi3,Fra}, the closed
expression in (\ref{derncot}) remained an open problem for several
years \cite{Apo,Bern,Hof,Knu,Kol}. The proof given here is worth
reporting, because, besides being an independent short proof of a
non-trivial problem \cite{Fra}, a natural interpretation of the
proof from the point of view of the approach addressed here is at
our disposal. Simple operations on the Zeon algebra (see
Definitions \ref{Def1}, \ref{Def2} and Lemma \ref{Lemma1} in
Section \ref{Basics}) and the representation of the ordinary
derivative and the Stirling numbers of the second kind as a
Berezin integral (see Lemma \ref{Lemma2} and Section \ref{proof})
comprise our main ingredients.

Although we will focus on the proof of Theorem \ref{thmcotsec},
our approach is also suitable to prove (following the steps
described in Section \ref{proof}) closed expressions for the
$n$-th derivative of tangent ($\tan$), cosecant ($\csc$), and the
hyperbolic analogues of all the functions cited.

Before we continue, we establish the basic underlying algebraic
setup needed to give the proof of Theorem \ref{thmcotsec}.

\section{Basics of the Zeon algebra and Berezin
integration}\label{Basics}

Let $\mathbb{C}$, $\mathbb{R}$, $\mathbb{Z}$ be the complex
numbers, real numbers, and integers, respectively.

\begin{definition}\label{Def1} The
\textit{Zeon algebra} $\mathcal{Z}_n \supset \mathbb{C}$ is
defined as the associative algebra generated by the collection
$\{\varepsilon_j\}_{j=1}^n$ ($n&lt;\infty$) and the scalar $1 \in
\mathbb{C}$, such that
$1\varepsilon_j=\varepsilon_j=\varepsilon_j1$, $\varepsilon_j
\varepsilon_k = \varepsilon_k \varepsilon_j$ $\forall$ $j$, $k$
and $\varepsilon_j^2=0$ $\forall$ $j$.
\end{definition} Note that
only linear elements in $\mathcal{Z}_n$ contribute to the
calculations.

For $\{j,k,\ldots,l\} \subset \{1,2,\ldots,n\}$ and
$\varepsilon_{jk\cdots l}\equiv \varepsilon_j\varepsilon_k\cdots
\varepsilon_l$ the most general element with $n$ generators
$\varepsilon_j$ can be written as (with the convention of sum over
repeated indices implicit)
\begin{equation}\label{phi}\phi_n=
a+a_j\varepsilon_j+a_{jk}\varepsilon_{jk}+\cdots+ a_{12\cdots
n}\varepsilon_{12\cdots n}=\sum_{\mathbf{j} \in
2^{[n]}}a_{\mathbf{j}}\varepsilon_\mathbf{j},\end{equation} with
$a$, $a_j$, $a_{jk}$, $\ldots$, $a_{12\cdots n}$ $\in$
$\mathbb{C}$, $2^{[n]}$ being the power set of
$[n]:=\{1,2,\ldots,n\}$, and $1\leq j&lt;k&lt; \cdots \leq n$. We refer
to $a$ as the body of $\phi_n$ and write $b(\phi_n)=a$ and to
$\phi_n-a$ as the soul such that $s(\phi_n)=\phi_n-a$. The
top-term is given by $\varepsilon_{12\cdots
n}=\varepsilon_1\varepsilon_2\cdots \varepsilon_n$, since
$\varepsilon_{12\cdots n}$ contains all the elements of
$\{\varepsilon_j\}_{j=1}^n$. Note that $\varepsilon_{12\cdots n}
\varepsilon_j = 0$ for all $j=1,\ldots,n$. In Lemma \ref{Lemma1}
we will also use the notation $\phi_n\equiv
\phi_n\bigl(\mbox{\boldmath $\varepsilon$}\equiv
\bigl(\varepsilon_1,\ldots,\varepsilon_n\bigr)\bigr)$ to indicate
directly the dependence of $\phi_n$ on the generators of the Zeon
algebra $\mathcal{Z}_n$.

Any sufficiently smooth function $f(z)$ in the complex domain
admits an extension to the context of the Zeon algebra from
previous results due to DeWitt \cite[Chapter 1]{DeWitt} and Rogers
\cite[Chapter 4]{Rogers}. More precisely, we have \cite[Equation
(1.1.6)]{DeWitt}
\begin{equation}\label{func}
f(\phi_n):=\sum_{j=0}^n\frac{f^{(j)}\bigl(b\bigl(\phi_n\bigr)\bigr)}{j!}s^j\bigl(\phi_n\bigr)
=\sum_{j=0}^n\frac{f^{(j)}\bigl(a\bigr)}{j!}s^j\bigl(\phi_n\bigr),
\end{equation} where $f^{(j)}(a)=d^jf(z)/dz^j|_{z=a}$ is the $j$-th ordinary derivative of $f(z)$ at $a$.
Note that $f\bigl(a+s\bigl(\phi_n\bigr)\bigr)|_{s(\phi_n)=0}=f(a)$
and $f\bigl(a+s\bigl(\phi_n\bigr)\bigr) \in \mathcal{Z}_n$,
because $s^{n+1}(\phi_n)=0$.

Concrete examples of (\ref{phi}), which will be important in the
proof of Theorem \ref{thmcotsec}, are the generalization of the
ordinary exponential and the generalized inverse ($a\equiv
b(\phi_n)\neq 0$ for the generalized inverse) given by
\begin{equation}\label{expinv}
e^{\phi_n}=
e^{a}\sum_{j=0}^n\frac{s^j\bigl(\phi_n\bigr)}{j!}\,\,{\rm and}\,\,
\phi_n^{-1}=\frac{1}{a}
\sum_{j=0}^n(-1)^j\displaystyle\frac{s^j(\phi_n)}{a^j},\end{equation}
respectively. Particular cases of (\ref{expinv}) are given by
$e^{1+\sqrt{2}\varepsilon_1-\varepsilon_3+i \varepsilon_{23}}$ $=$
$e\bigl(1-\varepsilon_3+ \sqrt{2}\varepsilon_1+i
\varepsilon_{23}\bigr.$ $\bigl.-\sqrt{2}\varepsilon_{13}+i
\sqrt{2} \varepsilon_{123}\bigr)$ with $n \geq 3$ and
$(1-\varepsilon_2+\varepsilon_{146})^{-1}$ $=$
$1+\varepsilon_2-\varepsilon_{146}-2\varepsilon_{1246}$ with $n
\geq 6$.

Using (\ref{expinv}) we can also define more complex functions,
which will be needed in this work, such as $\cot (\phi_n)$. The
generalization of the cotangent function is defined by
\begin{equation}\label{defcot}
\cot (\phi_n):=i\frac{e^{i \phi_n}+e^{-i \phi_n}}{e^{i \phi_n}-e^{-i \phi_n}}
\end{equation} with $b(\phi_n)\equiv a \in \mathbb{R}\backslash \{k\pi:k \in \mathbb{Z}\}$. Note that
$b\bigl(e^{i \phi_n}-e^{-i \phi_n}\bigr)=e^{i a}-e^{-i a}\neq 0$.
Therefore, $\cot (\phi_n)$ is well-defined using (\ref{expinv})
and, as expected, $\cot(\phi_n)|_{s(\phi_n)=0}=\cot(a)$.

\begin{definition}\label{Def2} The \textit{Berezin integral} on $\mathcal{Z}_n$, denoted
by $\int$, is the linear functional $\int: \mathcal{Z}_n
\rightarrow \mathbb{C}$ such that (we use throughout this work the
compact notation $d\mu_n:=d\varepsilon_n \cdots d\varepsilon_1$)
$$
d\varepsilon_jd\varepsilon_k=d\varepsilon_kd\varepsilon_j,\,\,
\int \phi_n\bigl(\hat{\varepsilon}_j\bigr)d\varepsilon_j=0\,\,{\rm and}
\int \phi_n\bigl(\hat{\varepsilon}_j\bigr)\varepsilon_jd\varepsilon_j=\phi_n\bigl(\hat{\varepsilon}_j\bigr),
$$ where $\phi_n\bigl(\hat{\varepsilon}_j\bigr)$ means any element of $\mathcal{Z}_n$ with no dependence on
$\varepsilon_j$. Multiple integrals are iterated integrals, i.e.,
\begin{equation}\label{Multi}
\int f(\phi_n) d\mu_n = \int \cdots \biggl(\int \biggl(\int f(\phi_n) d\varepsilon_n\biggr)
d \varepsilon_{n-1}\biggr)\cdots d\varepsilon_1.
\end{equation}\end{definition}
For example, it follows directly from Definition \ref{Def2} that $
\int d\varepsilon_j=0$, $\int \varepsilon_jd\varepsilon_j=1$,
$\int \varepsilon_j\varepsilon_kd\varepsilon_k =\varepsilon_j$
$\forall$ $j\neq k$ and $\int \phi_n d\mu_n = a_{12\cdots n}$. In
other words, the Berezin integral of $\phi_n$ in (\ref{phi}) gives
the coefficient of the top-term $\varepsilon_{12\cdots n}$. For
more details on Berezin integration we refer the reader to the
books of Berezin \cite[Chapter 1]{BreSQ} and \cite[Chapter
2]{BreIS}.

It is a direct consequence of Definition \ref{Def2} that a change
of variables formula holds for the Berezin integral. The proof of
the change of variables formula is routine and a detailed proof is
presented in, e.g., Rogers \cite[Theorem 11.2.3]{Rogers} or DeWitt
\cite[Chapter 1]{DeWitt} in the context of the Grassmann algebra
and can be straightforwardly adapted to $\mathcal{Z}_n$. For our
purposes it is sufficient to consider how a simple linear
transformation acting on the generators of $\mathcal{Z}_n$ affects
the Berezin integral. From now on $\per(A)$ means the permanent of
the matrix $A$.
\begin{lemma}\label{Lemma1} Let $\varepsilon_{j}'=a_{jk}\varepsilon_{k}$ with $a_{jk}\in
\mathbb{C}$, and let $A=\bigl[a_{jk}\bigr]$ be a square matrix of
order $n$ such that $\det(A)\neq 0$ and $\per(A^{-1})\neq 0$. With
the previous assumptions and notation the following formula holds
\begin{equation}\label{ChangeInt}
\int \phi_n(\mbox{\boldmath $\varepsilon$})d\mu_n
=\bigl({\rm per}\bigl(A^{-1}\bigr)\bigr)^{-1}\int \phi_n(A^{-1}\mbox{\boldmath $\varepsilon'$})d\mu_n'.
\end{equation}\end{lemma}

\begin{remark}
Note that $\bigl(\per\bigl(A^{-1}\bigr)\bigr)^{-1}$ instead of
$\bigl(\det (A)\bigr)^{-1}$ (as it occurs in ordinary calculus)
appears in (\ref{ChangeInt}).
\end{remark}

The representation of the ordinary derivative as a Berezin
integration can be traced back to previous papers
\cite{Bedi,Cara}. The proof of the aforementioned representation
follows by calculating the Berezin integral of both sides of
(\ref{func}) with $s(\phi_n) \equiv \varphi_n:=\sum_{j=1}^n
\varepsilon_j$ and observing that $\int \varphi_n^k
d\mu_n=k!\delta_{k,n}$ with $d\mu_n=d\varepsilon_n \cdots
d\varepsilon_1$. Finally, we obtain the desired result using the
multinomial theorem and Definition \ref{Def2}. For instance, see
\cite[Lemma 4.1 and Corollary 4.2]{Bedi}.
\begin{lemma}\label{Lemma2} Let $f$ be a sufficiently smooth function,
and let $\varphi_n=\sum_{j=1}^n \varepsilon_j$, where
$\{\varepsilon_j\}_{j=1}^n$ is the set of generators of the Zeon
algebra $\mathcal{Z}_{n}$. For $x\in\mathbb{C}$ and
$d\mu_n=d\varepsilon_n \cdots d\varepsilon_1$, the following
Berezin integral representation of the $n$-th ordinary derivative
of $f$ holds
$$
\int f(x+\varphi_n) d\mu_n=f^{(n)}(x).
$$
\end{lemma}

\section{Proof of Theorem \ref{thmcotsec}}\label{proof}

Before we prove Theorem \ref{thmcotsec} we will need some
auxiliary results.

It is a classical result in combinatorics that the following
generating function holds for the Stirling numbers of the second
kind \cite[Section 1.9]{Stan}:
$$
G_S(x)=\frac{1}{k!}\bigl(e^x-1\bigr)^k=\sum_{m=k}^{\infty}S(m,k)\frac{x^m}{m!}.
$$ Using (\ref{func})
and Lemma \ref{Lemma2} with $f=G_S$ we find
\begin{equation}\label{S2ndkind}
S(n,k)=\frac{1}{k!}\int \bigl(e^{\varphi_n}-1\bigr)^kd\mu_n,
\end{equation} where $e^{\varphi_n}$ is defined by (\ref{expinv}).
Setting $A_n\equiv
e^{\varphi_n}-1=\prod_{j=1}^ne^{\varepsilon_j}-1=\prod_{j=1}^n(1+\varepsilon_j)-1$
(using (\ref{expinv}) and $\varepsilon_j^2=0$) in (\ref{S2ndkind})
we get $S(n,k)=\int A_n^{k}d\mu_n/k!$, recovering the
representation of $S(n,k)$ introduced by Schott and Staples
\cite{Scho} and proved there directly from the definition of
$S(n,k)$ in terms of partitions of a finite set. In this way,
\cite[Definition 1.3]{Scho} is compatible with Definition
\ref{Def2}. Here we use a slightly different notation from that
adopted by Schott and Staples \cite{Scho}, with a subscript $n$
added to $A_n$.

Using the representation of the ordinary derivative in Lemma
\ref{Lemma2} we are ready to prove Theorem \ref{thmcotsec}.

\begin{proof}[Proof of Theorem \ref{thmcotsec}] Setting $f(x)=\cot (x)$ in Lemma \ref{Lemma2},
$x \in \mathbb{R}\backslash \{k\pi: k\in \mathbb{Z}\}$, we have
the representation
\begin{eqnarray}\label{begindem}
\cot^{(n)}(x)&amp;=&amp;\int \cot(x+\varphi_n) d\mu_n
= i\int\frac{e^{i\bigl(x+\varphi_n\bigr)}
+e^{-i\bigl(x+\varphi_n\bigr)}}
{e^{i\bigl(x+\varphi_n\bigr)}-e^{-i\bigl(x+\varphi_n\bigr)}}d\mu_n\vspace{2mm}\nonumber\\
&amp;=&amp;i\int\frac{e^{i\bigl(x+2\varphi_n\bigr)}+e^{-i x}}
{e^{i\bigl(x+2\varphi_n\bigr)}-e^{-i x}}d\mu_n
=i(2i)^n\int\frac{e^{i x+\varphi_n}+e^{-i x}}
{e^{i x+\varphi_n}-e^{-i x}}d\mu_n.
\end{eqnarray} Equation (\ref{begindem}) follows from (\ref{defcot}) and the change of variables formula of Lemma \ref{Lemma1}
with $a_{jk}=a_j\delta_{jk}$ and $a_j=2 i$ $\forall$ $j$. Note
that $\bigl({\rm per}\bigl(A^{-1}\bigr)\bigr)^{-1}= \bigl({\rm
per}[\delta_{jk}/a_j]\bigr)^{-1}=(2 i)^n$. Now we write
$e^{\varphi_n}= \bigl(e^{\varphi_n}-1\bigr)+1$ and use Euler's
formula $e^{i x}=\cos (x)+i\sin (x)$ to obtain
\begin{equation}\label{begindem0}
\frac{e^{i x+\varphi_n}+e^{-i x}}
{e^{i x+\varphi_n}-e^{-i x}}=
\frac{2\cos (x)+e^{i x}\bigl(e^{\varphi_n}-1\bigr)}{2i \sin (x)+e^{i x}\bigl(e^{\varphi_n}-1\bigr)}.
\end{equation}
For $k=0$ and $k&gt;n$, we have
\begin{equation}\label{begindem1}\int \bigl(e^{\varphi_n}-1\bigr)^kd\mu_n=0,\end{equation}
since $\int d\mu_n=0$ ($n \geq 1$) and
$$\bigl(e^{\varphi_n}-1\bigr)^{n+1}=
\varphi_n^{n+1}\biggl(\sum_{j=0}^{n-1}\frac{\varphi_n^j}{(j+1)!}\biggr)^{n+1}
=0.$$ Next, we use (\ref{expinv}) with $\phi_n\equiv 2i \sin
(x)+e^{i x}\bigl(e^{\varphi_n}-1\bigr)$, (\ref{begindem0}),
(\ref{begindem1}), and the linearity of the Berezin integral to
get
$$
\begin{array}{lll}
\int \frac{e^{i x+\varphi_n}+e^{-i x}}
{e^{i x+\varphi_n}-e^{-i x}}d\mu_n&amp;=&amp;
\sum_{k=0}^n(-1)^ke^{i k x}\int
\frac{\bigl(2\cos (x)+e^{i x}\bigl(e^{\varphi_n}-1\bigr)\bigr)\bigl(e^{\varphi_n}-1\bigr)^k}
{\bigl(2 i \sin (x)\bigr)^{k+1}}d\mu_n\\
&amp;=&amp;\bigl(\frac{\cos (x)}{i \sin (x)}-1\bigr)
\sum_{k=1}^n(-1)^ke^{i k x}\int
\frac{\bigl(e^{\varphi_n}-1\bigr)^k}
{\bigl(2 i \sin (x)\bigr)^k}d\mu_n\\
&amp;=&amp;-\bigl(i \cot (x)+1\bigr)
\sum_{k=1}^n\frac{k!}{2^k}\bigl(i \cot (x)-1\bigr)^k
\frac{1}{k!}\int\bigl(e^{\varphi_n}-1\bigr)^kd\mu_n.
\end{array}$$ Going back to (\ref{begindem}) and using the Berezin integral
representation of $S(n,k)$ of (\ref{S2ndkind}) we obtain the
desired result, i.e., Equation (\ref{derncot}).

Using Lemma \ref{Lemma2}, $x \in \mathbb{R}\backslash
\{(2k+1)\pi/2: k\in \mathbb{Z}\}$, we get
$$
\sec^{(n)}(x)=\int \sec(x+\varphi_n)d\mu_n
=\int \frac{2}{e^{i(x+\varphi_n)}+e^{-i(x+\varphi_n)}}d\mu_n
=\int \frac{2e^{i\varphi_n}}{e^{i(x+2\varphi_n)}+e^{-i x}}d\mu_n.
$$ Now we write $e^{2i \varphi_n}=(e^{2i \varphi_n}-1)+1$ and use (\ref{expinv}) to get
\begin{equation}\label{begindem2}
\sec^{(n)}(x)=2
\sum_{l=0}^n(-1)^le^{i lx}\displaystyle\int e^{i\varphi_n}
\frac{\bigl(e^{i 2\varphi_n}-1\bigr)^{l}}{\bigl(2 \cos (x)\bigr)^{l+1}}d\mu_n.
\end{equation} Next, we make the expansion $e^{i\varphi_n}=\prod_{j=1}^n(1+i \varepsilon_j)$
and, as a result, we need to analyze a general term such as
\begin{eqnarray}\label{begindem3}
\sum_{1\leq j_1&lt; \cdots &lt;j_k\leq n}i^{k}
\int \varepsilon_{j_1}
\cdots \varepsilon_{j_k}\bigl(e^{i
2\varphi_n}-1\bigr)^{l}d\mu_n&amp;=&amp;
{n \choose k}i^{k}\int \varepsilon_n
\cdots \varepsilon_{n-k+1}\bigl(e^{i
2\varphi_n}-1\bigr)^{l}d\mu_n\nonumber\\&amp;=&amp;
{n \choose k}i^{k}\int \bigl(e^{i
2\varphi_{n-k}}-1\bigr)^{l}d\mu_{n-k}\nonumber\\
&amp;=&amp;{n \choose k}(2i)^{n-k}i^{k}\int \bigl(e^{
\varphi_{n-k}}-1\bigr)^{l}d\mu_{n-k}.
\end{eqnarray} The invariance of $e^{i
2\varphi_n}$ under permutations of $\varepsilon_j$ with
$j=1,\ldots,n$ was used to obtain the first equality. The second
equality follows from
$$\varepsilon_n\cdots \varepsilon_{n-k+1}\bigl(e^{i
2\varphi_n}-1\bigr)^{l}=\varepsilon_n\cdots
\varepsilon_{n-k+1}\bigl(e^{i 2\varphi_{n-k}}-1\bigr)^{l}$$ and
(\ref{Multi}). Finally, the change of variables formula of Lemma
\ref{Lemma1} was used to obtain the last equality. Note that the
constraint $n-k \geq l$ follows from the properties of the Berezin
integral. Using (\ref{begindem3}) and the representation of
(\ref{S2ndkind}) in (\ref{begindem2}) we obtain the desired
result, i.e., Equation (\ref{dernsec}).
%\qed
\end{proof}

\section{Concluding remarks}

We have shown that closed formulas for the $n$-th derivative of
the cotangent and secant functions in Theorem \ref{thmcotsec}
follow from simple computations in the context of the Zeon
algebra. Our approach is also suitable to give closed formulas for
higher order derivatives of other trigonometric functions, i.e.,
$\csc$, $\tan$ and hyperbolic functions such as $\coth$, $\sech$,
$\csch$ and $\tanh$. Our starting point was an extension of a
function in the complex domain to the more general scenario of the
Zeon algebra (see (\ref{func})). Along the way, the Berezin
integral representation of the Stirling numbers of the second kind
played a key role in our analysis. The aforementioned extension
led us to prove Theorem \ref{thmcotsec} quite naturally using
known results about the Zeon algebra (Lemma \ref{Lemma1} and Lemma
\ref{Lemma2}) and, at the same time, taking advantage of the
computational power of the Zeon algebra, i.e., the fact that only
linear terms on the generators appear in the calculations. The
final message is that techniques based on super-analysis
\cite{BreSQ,BreIS}, as it occurs in other contexts
\cite{Abde,Cara,Mansour,Schor}, may provide a useful computational
toolbox in representing combinatorial numbers, such as the
Stirling numbers of the second kind, and in proving combinatorial
identities of the type considered here.

\section{Acknowledgments}
The author thanks the anonymous referee 
for suggestions that improved the paper.

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\end{thebibliography}

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\noindent 2010 {\it Mathematics Subject Classification}: Primary
11B73; Secondary 33B10; 05A15;  05A18;  05A19.

\noindent \emph{Keywords: } Zeon algebra, Berezin integration,
cotangent, secant, Stirling number of the second kind, generating
function.

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\noindent (Concerned with sequence \seqnum{A008277}).

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\vspace*{+.1in}
\noindent
Received
April 13 2014;
revised versions received
July 8 2014; July 16 2014; August 6 2014.
Published in {\it
Journal of Integer Sequences}, August 12 2014.


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